GCD Monoid
(monoid/gcd.hpp)

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• Last update: 2026-07-21 20:17:47+09:00
• Include: #include "monoid/gcd.hpp"

Overview

A monoid representing the Greatest Common Divisor (GCD) operation. This is an idempotent monoid, meaning it can be safely used with a Sparse Table for $O(1)$ Range GCD queries.

Template Parameters

• T: The underlying integer data type.

Properties

• Operation: std::gcd(a, b)
• Identity Element: $0$ (Since $\gcd(x, 0) = x$)

Interface and Complexity

This is a stateless algebra tag. Generic data structures use its public value_type, id(), and op(a, b) members. If the type also provides helpers such as make(...) or inv(x), they are described above or in the documented properties.

Each static operation runs in the cost of the underlying operation shown in the properties. Scalar monoids are $O(1)$; monoids whose value_type stores several items, permutations, or matrices scale with that stored size.

Verified with

• verify/monoid/commutative_flags.test.cpp

Code

#ifndef M1UNE_MONOID_GCD_HPP
#define M1UNE_MONOID_GCD_HPP 1

#include <numeric>

namespace m1une {
namespace monoid {

// Monoid for Greatest Common Divisor (Range GCD).
template <typename T>
struct Gcd {
    using value_type = T;
    static constexpr bool commutative = true;

    // The identity element for GCD is 0.
    static constexpr T id() {
        return T(0);
    }

    // Returns the greatest common divisor of a and b.
    static constexpr T op(const T& a, const T& b) {
        return std::gcd(a, b);
    }
};

}  // namespace monoid
}  // namespace m1une

#endif  // M1UNE_MONOID_GCD_HPP

#line 1 "monoid/gcd.hpp"



#include <numeric>

namespace m1une {
namespace monoid {

// Monoid for Greatest Common Divisor (Range GCD).
template <typename T>
struct Gcd {
    using value_type = T;
    static constexpr bool commutative = true;

    // The identity element for GCD is 0.
    static constexpr T id() {
        return T(0);
    }

    // Returns the greatest common divisor of a and b.
    static constexpr T op(const T& a, const T& b) {
        return std::gcd(a, b);
    }
};

}  // namespace monoid
}  // namespace m1une

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